A Construction of Codebooks Associated With Binary Sequences
A Construction of Codebooks Associated With Binary Sequences
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DOI：
10.1109/tit.2012.2196021
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发表时间：
2012-08
影响因子：
2.5
通讯作者：
N. Yu
中科院分区：
文献类型：
--
作者：
N. Yu
（N，K）码本是K维向量空间中的N个单位范数码向量的集合。对于其应用，期望一对不同码向量之间的内积的最大幅度应尽可能小，严格或渐近地满足Welch界等式。本文从一个K × N部分矩阵构造（N，K）码书，其中K <; N，每个码向量等价于矩阵的一列。为了获得K × N矩阵，从与长度为J和汉明权重为K的二进制序列相关联的J × N矩阵Φ中选择K行，其中所选择的行索引的集合等价于二进制序列的非零条目的索引集合。然后发现，一对不同码向量之间的内积的最大幅度由二进制序列的Φ-变换的最大幅度决定。因此，构造具有小幅度的内积的码本等效于找到其中其Φ变换的最大幅度尽可能小的二进制序列。从发现，新的类的码书与非平凡的界限上的最大内积构造傅立叶和Hadamard矩阵与二进制序列。
An (N, K) codebook is a set of N unit-norm code vectors in a K-dimensional vector space. For its applications, it is desired that the maximum magnitude of inner products between a pair of distinct code vectors should be as small as possible, meeting the Welch bound equality strictly or asymptotically. In this paper, an (N, K) codebook is constructed from a K × N partial matrix with K <; N, where each code vector is equivalent to a column of the matrix. To obtain the K × N matrix, K rows are selected from a J × N matrix Φ, associated with a binary sequence of length J and Hamming weight K, where a set of the selected row indices is equivalent to the index set of nonzero entries of the binary sequence. It is then discovered that the maximum magnitude of inner products between a pair of distinct code vectors is determined by the maximum magnitude of Φ-transform of the binary sequence. Thus, constructing a codebook with small magnitude of inner products is equivalent to finding a binary sequence where the maximum magnitude of its Φ-transform is as small as possible. From the discovery, new classes of codebooks with nontrivial bounds on the maximum inner products are constructed from Fourier and Hadamard matrices associated with binary sequences.